The degree of the Jacobian locus and the Schottky problem

Details The degree of the Jacobian locus and the Schottky problem The degree of the Jacobian locus and the Schottky problem

2004-02-29

arxiv.org/abs/math/0403009v1

Mathematics

Algebraic Geometry

Scientific paper

We show that the degree of the images of the moduli space of (principally polarized) abelian varieties A_g and of the moduli space of curves M_g in the projective space under the theta constant embedding are equal to the top self-intersection numbers of one half the first Hodge class on them. This allows us to obtain an explicit formula for the degree of A_g, and an explicit upper bound for the degree of M_g. Knowing the degree of A_g allows us to effectively determine the subvariety itself, i.e. to effectively obtain all polynomial equations satisfied by theta constants. Furthermore, combining the bound on the degree of M_g with effective Nullstellensatz allows us to rewrite the Kadomtsev-Petvsiashvili (KP) partial differential equation as a system of algebraic equations for theta constants, and thus obtain an effective algebraic solution to the Schottky problem.

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